INTRODUCTION - University of Malta
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PROGRAM OF STUDIES IN MATHEMATICS FOR THE B.Sc. (Hons.) degree ACADEMIC YEAR 2007-2008 25th May 2007. Year I starting 2007-2008 Mathematics students in the first year of the B.Sc.(Hons.) course must take all the compulsory credits and a selection of at most 8 optional credits. Type of unit Code Title of unit Credit Value Semester Lecturers 3 1st 5 1st Dr. J. L. Borg Dr. E. Chetcuti Compulsory Credits: MAT1090 Mathematical Methods Prof. I. Sciriha Dr. A. Vella MAT1000 Introductory Mathematics MAT1511 Analytical Geometry 4 1st MAT1101 Groups and Vector Spaces 6 2nd Prof. I. Sciriha 4 2nd Dr. D. Buhagiar 4 2nd Prof. J. Lauri MAT1211 MAT1411 Analysis I Discrete Methods Optional Credits: A total of 8 credits may be chosen from units designated as such in any program of study within the University. The following study unit is highly recommended, especially for students who do not have Advanced Applied Mathematics: MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti The following study unit/s are highly recommended, especially for students who have not done probability or statistics. & SOR1110 OR & SOR1211 & SOR1221 & Probability 4 1st & 2nd Probability and Sampling, Estimation and Regression 2 2 1st 2nd Dr. L. Sant Various Various These credits cannot be taken by students taking Statistics and Operations Research as a Principal subject area for the B.Sc. (Hons.). The credits SOR1211 and SOR1221 are slightly easier than SOR1110, and cannot be taken with it. Other optional credits can be chosen from: 1 † PHY1020 or/and † PHY1030 Basic Concepts in Physics I 2 1st Basic Concepts in Physics II 2 2nd # Programming for Scientists 2 1st Linear Programming 2 2nd Principles and Perspectives of Science 2 2nd CIS1003 or/and £ SOR1311 or/and ~ CHE2060 † Not to be taken by students taking Physics as a principal subject area. # Not to be taken by students taking Computer Information Systems or Computer Science and AI as a principal subject area. £ Not to be taken by students taking Statistics as a principal subject area. ~ Not to be taken by students taking Chemistry as a principal subject area. Optional credits offered to other departments The Mathematics department offers the following as optional credits to other departments and/or faculties: # MAT1001 Mathematical Methods I: (i) Matrices 2 1st 2 st Prof. I. Sciriha / Mr. J. Borg # MAT1002 Mathematical Methods I: (ii) Ordinary 1 Differential Equations # MAT1091 Mathematical Methods I: Matrices and Dr. A. Vella / Ms N. Camillleri st 4 1 Various Differential Equations % MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti & MAT1612 Applied Mathematics for Engineers 4 1st Mr. E. Cardona $ MAT1900 Elementary Calculus 4 1st Ms G. Galea # Students can only register for one study unit from MAT1001, MAT1002 and MAT1091. These credits cannot be taken by students in B.E.& A., B.Eng., and B.Sc. with Mathematics as one of the options. % Given twice in the first semester, once to B.Sc.(Hons.) and B.E.& A. students, and once to B.Eng. and B.Ed.(Hons.) students. & For B.Eng. and B.E.& A. students only. $ Cannot be taken by students in B.Com., B.E.& A., B.Eng. or B.Ed. (Mathematics and/or Physics) students. Year II starting 2007-2008 2 Mathematics students in the second year of the B.Sc.(Hons.) course must take all the compulsory credits and a choice of four elective credits from the following list: Type of unit Code Title of unit Credit Semester Lecturers Value Compulsory Credits: MAT2112 MAT2212 MAT2412 Linear Algebra I Analysis II Introduction to Graph Theory 4 1st Prof. I. Sciriha 4 1st Dr. A. Vella 2 1s t Prof. J. Lauri Dr. J. Sultana MAT2512 Vector Analysis I 4 1st MAT2113 Rings 4 2nd Dr. A. Vella 4 2nd Dr. J. L. Borg 4 2nd Dr. J. Sultana MAT2213 MAT2513 Analysis III Vector Analysis II Elective Credits: Students must choose 4 credits from the following list. £ £ MAT2912 Computational Mathematics 2 1st & 2nd Various Elective credit offered only to students in the B.Sc. (Hons.) course. This credit is highly recommended to Mathematics students taking Statistics or Physics in the B.Sc.(Hons.) course. Other elective credits can be chosen from: * MAT1611 Introductory Mechanics 4 1st Mr. J. Borg / Mr. V. Cilia Vincenti ~ MAT2402 Networks 2 1st Prof. J. Lauri ^ PHY2150 Numerical Analysis 2 2nd Prof. A. Buhagiar 2 nd & CHE2060 Principles and perspectives of 2 Prof. F. Ventura Science SCI1200 Science of the Earth 4 1st & 2nd Dr. P. Galea / Mr. J. M. Bonnici st % SOR1211 Probability 2 1 Various % SOR1221 Sampling, Estimation and 2 2nd Various %† SOR1231 2 2nd Various % SOR1311 Regression Hypothesis testing and modeling using SPSS % SOR 1110 Linear Programming Probability 2nd 2 4 st Mrs. N. Attard nd 1 &2 Dr. L. Sant * Optional credit offered also to other departments. This credit is given twice in the first semester, once to B.Sc.(Hons.) and B.E.& A. students, and once to B.Eng. and B.Ed.(Hons.) students. ~ This credit is also offered to B.Sc. (I.T.) students. ^ Not to be taken by students taking Physics as a principal subject area. & Not to be taken by students taking Chemistry as a principal subject area. % Not to be taken by students taking Statistics and Operations Research as a principal subject area. † This study unit needs as prerequisites the credits SOR1211 and SOR1221. Year III 2007- 2008 only 3 Mathematics students in the third year of the B.Sc. (Hons.) course must take the following credits, all of which are compulsory: Type of unit Code Title of unit Credit Semester Lecturers Value Compulsory credits: MAT3105 MAT3214 MAT3905 MAT3792 Linear Algebra I Complex Analysis Linear Algebra II Metric Spaces II & 4 1st Prof . I. Sciriha 4 1st Dr. J. Sultana 2 1st Prof. I. Sciriha 4 1st Dr. D. Buhagiar Uniform Convergence Year III & Dr. J. L. Borg MAT3206 Lebesgue Integration 4 2nd Dr. D. Buhagiar MAT3504 Vector Analysis IIA 4 2nd Dr. J. Sultana MAT3602 Mechanics 4 2nd Prof. A. Buhagiar MAT3701 Differential Equations 4 2nd Prof. A. Buhagiar starting in 2008-2009 Mathematics students in the third year of the B.Sc. (Hons.) course must take the following credits, all of which are compulsory: Type of unit Code Title of unit Credit Semester Lecturers Value Compulsory credits: MAT3114 MAT3115 MAT3214 MAT3215 Groups Complex Analysis Metric Spaces 2 1st Prof. I. Sciriha 4 1st Prof. J. Lauri 4 1st Dr. J. Sultana 4 1st Dr. D. Buhagiar Dr. A. Vella MAT3413 Probabilistic Combinatorics 2 1st MAT3216 Analysis IV 2 2nd Dr. J. L. Borg 4 2nd Dr. D. Buhagiar 4 2nd Prof. A. Buhagiar 4 2nd Dr. J. Muscat MAT3217 MAT3612 MAT3711 Year IV Linear Algebra II Lebesgue Integration Mechanics Differential Equations starting 2007-2008 4 One credit in the fourth year of the B.Sc. course is equivalent to five hours of lectures and/or tutorials. Students taking 34 credits in Mathematics must take: exactly 18 credits from those labelled Project, the compulsory credit MAT3207 Complex Analysis, one elective of 5 credits, and one elective of 6 credits. Students taking 26 credits in Mathematics must take: the compulsory credits MAT3990 Mathematics Seminar and MAT3207 Complex Analysis, together with a choice of three electives of 5 credits each. Type of unit Code Title of unit Credit Semester Lecturers Value Project MAT3218 Functional Analysis 12 1st & 2nd Dr. D. Buhagiar / Dr. J. Hamhalter Project MAT3414 Discrete Mathematics 12 1st & 2nd Prof. J. Lauri / Prof. I. Sciriha Project MAT3990 Mathematics Seminar 6 1st & 2nd Various Compulsory * MAT3207 Complex Analysis 5 1st & 2nd Dr. J. Sultana Elective ! MAT3613 Classical Mechanics 5 1st & 2nd Prof. A. Buhagiar Elective ! MAT3116 Group Representations 5 1st & 2nd Prof. J. Lauri Elective ! MAT3219 General Topology 5 1st & 2nd Dr. D. Buhagiar Elective !^ MAT3712 Partial 5 2nd 5 1st & 2nd Dr. J. Sultana 2nd Prof. A. Buhagiar Differential Equations & Dr. J. Muscat Calculus of Variations Elective ! MAT3513 Tensors and Relativity Elective ! MAT3713 The Finite Element Method 5 1st Elective ! MAT3693 Classical Mechanics 6 1st & 2nd Prof. A. Buhagiar 6 1st & 2nd Prof. J. Lauri 6 1st & 2nd Dr. D. Buhagiar Elective ! MAT3196 Group Representations Elective ! MAT3299 General Topology Elective !^ MAT3782 Partial Differential Equations & & 6 2nd 6 1st & 2nd Dr. J. Sultana 6 1st Prof. A. Buhagiar Dr. J. Muscat Calculus of Variations Elective Elective ! ! MAT3593 MAT3793 Tensors and Relativity The Finite Element Method & 2nd * Offered for the last time in the fourth year; subsequently offered only in the third year. ^ Not offered in 2007-2008, as the fourth years had already covered this in the third year. ! MAT3613 cannot be taken with MAT3693, nor MAT3116 with MAT3196, nor MAT3219 with MAT3299, nor MAT3712 with MAT3782, nor MAT3513 with MAT3593, nor MAT3713 with MAT3793. STUDY UNITS TO OTHER FACULTIES 5 The Department of Mathematics also gives credits in Mathematics to students in other faculties. Optional credits in Mathematics can be taken by students in other departments. Besides, B.Ed.(Hons.) students majoring in Mathematics can choose to follow most study units in the first two years of the B.Sc.(Hons.) course in Mathematics. The following study units are given to students in the Faculty of Engineering, FEMA and also to students reading for a B.Sc. degree in Information Technology. A detailed description of these courses is given in the following pages. Code Title of unit Credit Value Semester Lecturers Faculties of Engineering and Architecture 1st Year Optional credits MAT1611 Introductory Mechanics or MAT1612 Applied Mathematics for Engineers 4 1 st 4 1st 1st Year Core credits MAT1801 Mathematics for Engineers I 4 1st MAT1802 Mathematics for Engineers II 4 2nd Laplace and Fourier Transforms The Eigenvalue problem and multiple integrals 2 2 1st 2nd Dr. J. L. Borg Dr. J. L. Borg Faculty of Engineering 3rd Year MAT3805 Optimisation, Complex & Numerical Analysis I 4 1st MAT3806 2nd Dr. J. Sultana / Prof. A. Buhagiar Dr. J. Sultana / Prof. A. Buhagiar 2nd Year MAT2803 MAT2804 Optimisation, Complex & Numerical Analysis II 4 Faculty of Economics, Management and Accountancy (FEMA) 1st Year †MAT1991 Mathematics I and II 8 This study unit comprises two credits as follows: %† MAT1901 Mathematics I and † MAT1902 Mathematics II Board of Studies for Information Technology 1st Year *^ MAT1091 Mathematics Methods I: Matrices and Differential Equations MAT1803 Mathematical Transform Techniques 2nd Year *MAT1401 MAT2402 Discrete Methods Networks Mr.V.Cilia Vincenti Mr..E. Cardona Dr. J. L. Borg / Various Dr. A. Vella / Various 1st & 2nd Prof. A. Buhagiar/ Various 4 1st 4 2nd Dr. A. Vella / Various Mr. J. Sciriha / Various 4 1st 2 2nd Prof . I. Sciriha / Various Mr. J. Borg 4 2 2nd 1st Prof. J. Lauri Prof. J. Lauri * Students in I.T. attend these Mathematics courses together with students reading Mathematics in the B.Sc. (Hons.) degree. The description of these courses is already given under the Mathematics option in the B.Sc. (Hons.) degree. ^ Amalgamated version of the credits MAT1001 and MAT1002; these are still individually available. % For students in B.Sc. (Tourism Studies) † Students cannot take MAT1991 with MAT1901 and/or MAT1902. Brief information on the Department of Mathematics. 6 Mathematics underlies the pursuit of every scientific endeavour. The Department of Mathematics within the Faculty of Science satisfies this need by contributing to joint Honours degrees with other science disciplines such as Physics, Statistics and Computer Science. Besides it also gives courses to other Faculties, like Education, Engineering, Architecture, FEMA, and I.T. The following is a list of the Full Time Mathematics Academic Staff in the Department of Mathematics: • • • • • • • • Dr. A. Vella Dr. J. Sultana Prof. I. Sciriha Dr. M. Micallef Prof. J. Lauri Dr. D. Buhagiar Prof. A. Buhagiar, Dr. J. L. Borg Head of Department The number of students taking Mathematics as a principal subject for their B.Sc.(Hons.) has now stabilised at about 30 per year. Besides, about 40 students from the Faculty of Education annually follow the same Mathematics courses followed by the Mathematics students in the Faculty of Science. Student numbers in Mathematics courses for other Faculties vary greatly from faculty to faculty, with about 400 first year students in FEMA, to Engineering classes of about 100 to 200. The optional credits offered by the Mathematics Department are also very popular with students from other courses, with audiences of about 100. Part time staff are employed by the Department of Mathematics to give tutorials in Mathematics to small classes. The Department of Mathematics also offers postgraduate (Masters’) courses in Mathematics. Departmental research centres mainly round Graph Theory and Combinatorics, and Mathematical Physics. Except where otherwise stated, study units are assessed by written test according to the following norms: Year of Course Number of Credits Exam Duration (Hours) Number of Lecturers Number of Number of Questions Questions Set Chosen -----------------------------------------------------------------------------------------------------------------------------------------------------------I-III 2 11/2 - 3 2 I-III 4 2 1 2 4 4 3 3 I-III 6 3 1 2 3 7 8 9 5 5 5 IV 5 3 . 5 4 IV 6 3 - 6 5 IV Project 12 3 - 8 5 Masters 18 3 - 6 3 ------------------------------------------------------------------------------------------------------------------------------------------------------------ The study units given by the Department of Mathematics are described below. COURSE DESCRIPTIONS B.Sc(Hons.) 7 Year I MAT1090 - Mathematical Methods Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • Prof. I. Sciriha, Dr. A. Vella A-Level MAT1101, MAT3701 3 15 6 1 Matrices; Determinants; Solution of linear equations; Eigenvalues and diagonalisation. • Ordinary differential equations of the first order; • Ordinary differential equations of the second order; • Partial differentiation and exact differential equations. Method of Assessment: Examination 100% Textbooks • • • • Gow M., A Course in Pure Mathematics, Arnold, 1960. Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999. Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998. Derrick W. and Grossman S., Elementary Differential Equations, Addison-Wesley, 4th Edition, 1997. MAT1000 - Introductory Mathematics 8 Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. J. L. Borg A-Level MAT1101, MAT1211 5 25 10 1 • Natural numbers and the principle of induction. • Integers: • Divisibility, • Prime numbers, • The greatest common divisor and the Euclidean algorithm, • The fundamental theorem of arithmetic; • Rational and real numbers: • The completeness axiom, • The Archimedean property of the real numbers, • Inequalities. • Sets: • Inclusion, union, intersection, • De Morgan’s laws; • Ordered pairs and the Cartesian product of sets; • Relations: • Basic properties, • Equivalence relations and partitions of a set, • Functions: • The function as a mapping, • Injectivity and surjectivity, • Composition of functions, • Inverse functions; • Cardinality and Countability. Method of assessment: Examination 100%. Suggested Readings • D’Angelo J.P. and West D.B., Mathematical Thinking: Problem-Solving and Proofs, Prentice Hall, 2nd Edition, 2000. • Schumacher C., Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley, 1996. • Devlin K.J., Sets, Functions and Logic, Chapman and Hall, 1981. • Epp S., Discrete Mathematics with Applications, Brooks Cole, 1995. MAT1101 - Groups and Vector Spaces 9 Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • • • • • Prof. I. Sciriha MAT1000 MAT2112 6 30 12 2 Introduction to groups: symmetry, axiomatic approach; Lagrange's theorem; An introduction to number theory; Permutations; Normal subgroups; Quotient groups; First isomorphism theorem; Introduction to vector spaces; Linear transformations and matrices; Dimension theorem, nullity, rank; Change of basis. Method of Assessment: Examination 100% Main Texts • • • • Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. Cameron P., Introduction to Algebra, Oxford University Press, Oxford, 1998. Lipschutz S., Linear Algebra, Schaum’s Outline Series, McGraw-Hill, 2nd Edition, 1991. Wallace D., Groups, Rings and Fields, Springer, 1998. Supplementary Readings • Armstrong M.A., Groups and Symmetry, Springer Verlag, Heidelberg, 1997. • Anton H., Elementary Linear Algebra, John Wiley & Sons, 8th Edition, 2000. MAT1211 - Analysis I Lecturer: Dr. D. Buhagiar 10 Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: MAT1000 MAT2212 4 20 8 2 • The real number line: • Open and closed sets, • Compact sets; • Sequences of real numbers: • Convergence, • Basic theorems on limits of sequences, • Lim sup and lim inf, • The Bolzano-Weierstrass theorem, • The Cauchy convergence criterion; • Sequences in Rn: • The norm of a vector, • Convergence; • Series in R: • Conditional and absolute convergence, • Tests for convergence. Method of Assessment: Examination 100% Suggested reading: • • • • Spivak M,, Calculus, Publish or Perish, 3rd Edition, 1994 Abbott S., Understanding Analysis, Springer, 2001 Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999 Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974 MAT1411 - Discrete Methods Lecturer: Prof. J. Lauri 11 Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • A-Level MAT2412 4 20 8 2 Permutations, Combinations, Partitions of a set, The inclusion-exclusion principle; • Recurrence relations, • Generating functions, • Partitions of a positive integer. Method of Assessment: Examination 100% Textbooks: One of • Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 2nd Edition, 2002. or • Dossey J. A., Otto A. D., Spence L.E. and Eynden C.V., Discrete Mathematics, Addison–Wesley, 5th Edition, 2006. MAT1511 - Analytical Geometry Lecturer: Dr. E. Chetcuti Follows from: A-Level 12 Leads to: Credit value: Lectures: Tutorials: Semester: MAT2512 4 20 8 1 • Vector geometry: • Affine spaces, • Position vectors; • Euclidean geometry: • The dot product, • Angles, • Isometries; • Coordinates and equations: • Cartesian coordinates, • Curves and equations, • Coordinate form of an isometry, • Change of coordinates; • Orientation and vector product: • Vector algebra, • Vector equations of lines and planes; • Conics: • The ellipse, • The parabola, • The hyperbola. Method of Assessment: Examination 100% Textbooks • Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997. Supplemaentary Reading: • Vaisman I., Analytical Geometry, World Scientific Publishing Company, 1998. • Camilleri C.J., Vector Analysis, Malta University Press, Malta, 1994. MAT1611 – Introductory Mechanics Lecturers: Mr. J. Borg; Mr. V. Cilia Vincenti Follows from: Advanced Level Pure Mathematics 13 Leads to: Credit value: Lectures: Tutorials: Semester: MAT3602 4 20 8 repeated twice in Semester 1 • Vectors: • Distance, velocity and acceleration, • Force and Newton’s laws, • Components of a system of forces and their resultant, • Equilibrium and acceleration under concurrent forces; • Statics: • Systems in equilibrium, • Lami’s theorem, • Friction, • Centroids • Light frameworks, • Shearing force and bending moment in beams; • Dynamics: • Projectiles, • Circular motion, • Work and energy, • Elasticity, • Simple harmonic motion, • Momentum and impulse. Method of Assessment: Examination 100% Textbooks • Jefferson B. and Beadsworth T., Introducing Mechanics, Oxford University Press, 2000. • Jefferson B. and Beadsworth T., Further Mechanics, Oxford University Press, 2001. • Bostock L. and Chandler S., Applied Mathematics, Volumes 1 and 2, Stanley Thornes, 1975. Students who have Mathematics as a principal subject area in the B.Sc. course cannot take the following study units. MAT1612 – Applied Mathematics for Engineers 14 Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Mr. E. Cardona Advanced Level Pure Mathematics MAT1802 4 20 8 1 • Vectors: • Addition and subtraction of vectors, • Resolution of a vector, • Rectangular resolution of a vector; • Geometrical applications: • Vector equation of a straight line, • Vector equation of a plane; • Products of vectors: • Dot product, • Cross product, • Scalar triple product, • Vector triple product; • Differentiation and integration of vectors: • Derivative of a vector, • Derivative of products, • Integration, • Serret-Frenet formulae of curves: • Curvature, tangent, normal, binormal; • Coordinate geometry: • Equations of lines and curves, • The circle, the ellipse and the hyperbola; • Partial differentiation: • Applications; • Integration and applications: • First and second moment of area, • Centroid and centre of mass, • Mass moment of inertia. Method of Assessment: Examination 100% Suggested Reading: • Bostock L. and Chandler S., Applied Mathematics, Volumes 1 and 2, Stanley Thornes, 1975. • Jefferson B. and Beadsworth T., Introducing Mechanics, Oxford University Press, 2000. • Jefferson B. and Beadsworth T., Further Mechanics, Oxford University Press, 2001. MAT1001 - Mathematical Methods I: (i) Matrices Lecturers: Various Follows from: A-Level Leads to: MAT1101 15 Credit value: Lectures: Tutorials: Semester: • • • • 2 10 4 1 Matrices; Determinants; Solution of linear equations; Eigenvalues and diagonalisation. Method of Assessment: Examination 100% Textbooks • Gow M., A Course in Pure Mathematics, Arnold, 1960. • Andvilli S. and Hecker D., Elementary Linear Algebra, Harcourt Academic Press, 1999. MAT1002 - Mathematical Methods I: (ii) Ordinary Differential Equations Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Various A-Level MAT3701 2 10 4 1 • Ordinary differential equations of the first order; • Ordinary differential equations of the second order with constant coefficients; • Partial differentiation and exact differential equations. Method of Assessment: Examination 100% Textbooks • Gow M., A Course in Pure Mathematics, Arnold, 1960. • Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 8th Edition, 1998. • Derrick W. and Grossman S., Elementary Differential Equations, Addison-Wesley, 4th Edition, 1997. MAT1091 - Mathematical Methods I: Matrices and Differential Equations Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Various A-Level MAT1101, MAT3701 4 20 8 1 Method of Assessment: Examination 100% This credit is made up of the two study units MAT1001 and MAT1002 described above. MAT1900 - Elementary Calculus Lecturer: Ms. G. Galea 16 Follows from: Credit value: Lectures: Tutorials: Semesters: • • • • • • O-Level 4 20 8 1, 2 Cartesian coordinates; Equations of lines and curves; Coordinate geometry of the circle; Differentiation; Partial differentiation; Integration and applications. Method of Assessment: Examination 100% Textbooks • Bostock L. and Chandler S., The Core Course for A-Level, Stanley Thornes, The Bath Press, Avon, 1990. • Shepperd J.A.H. and Shepperd C.J., Pure Maths for A level, Hodder & Stoughton, 1983. Year II MAT2112 - Linear Algebra I 17 This credit is identical to MAT3105 in the third year. Second and third year students attend together for MAT2112 and/or MAT3105, and they will have the same examination. After 2008, MAT3105 will be discontinued, and only MAT2112 will be given to the second years only. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • • • • • • • • • • • Prof. I. Sciriha MAT1101 MAT3905 4 20 8 1 Vector spaces; Dimension theorems; Third isomorphism theorem; Direct sum of spaces; Dual spaces; Inner product spaces; Gram Schmidt orthogonalisation; The rank of matrices; Projection of a vector space onto a subspace; Similar matrices; Eigenvalues and eigenvectors; The characteristic polynomial; The Cayley-Hamilton theorem; The minimum polynomial; Diagonalisation and applications; The Jordan normal form; Spectral decomposition. Method of Assessment: Examination 100% Textbooks • • • • • Lecture Notes. Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998. Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. Nering E.D., Linear Algebra and Matrix Theory, John Wiley and Sons, 2nd Edition, 1970. Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003. MAT2113 - Rings Lecturer: Follows from: Leads to: Credit value: Lectures: Dr. A. Vella MAT1101 MAT3905 4 20 18 Tutorials: Semester: • • • • • • • • • 8 2 Axioms, examples of and elementary results on rings; Division rings, integral domains and fields; Ideals, quotient rings; Factorisation in rings; Prime and irreducible elements; Euclidean rings; Unique factorisation; Applications to some simple results in number theory; The ring of polymomials. Method of Assessment: Examination 100% Textbooks • Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. • Cameron P. J., Introduction to Algebra, Oxford University Press, 1998. MAT2212 Analysis II Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. A. Vella MAT1211 MAT2213 4 20 8 1 • Limits of functions: • Continuity, • The intermediate value theorem, • Continuous functions on [a, b], • Uniform continuity; • Differentiability and the derivative: • The mean value theorem, • Classification of critical points, • L’Hospital’s rules, • Taylor’s theorem. Method of Assessment: Examination 100% Suggested reading: • • • • Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994 Abbott S., Understanding Analysis, Springer, 2001 Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999 Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974 MAT2213 Analysis III Lecturer: Follows from: Leads to: Credit value: Lectures: Dr. J. L. Borg MAT2212 MAT3214, MAT3216 4 20 19 Tutorials: Semester: 8 2 • Riemann integration: • The Riemann integral, • Basic theorems on integrals, • The fundamental theorem of calculus, • Taylor’s theorem in integral form; • Uniform convergence: • The Weierstrass approximation theorem, • Applications to power series and Fourier series. Method of Assessment: Examination 100% Suggested reading: • • • • Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994 Abbott S., Understanding Analysis, Springer, 2001 Bartle R. & Sherbert D., Introduction to Real Analysis, Wiley, 3rd Ed., 1999 Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974 MAT2412 – Introduction to Graph Theory Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. J. Lauri MAT1411 MAT3414 2 10 4 1 • • • • • • • • Definitions, examples and elementary results; Trees; Connectivity, Max-Flow Min-Cut theorem, Menger's theorem; Euler tours and Hamilton cycles; Vector spaces associated with the edge-set of a graph; Planarity I: Euler's formula, K5 and K3,3, Kuratowski's theorem; Planarity II: the combinatorial dual, Maclane's characterization of planarity; Planarity III: the five colour theorem, the four colour theorem, and edge colouring of cubic graphs; • Colouring of graphs on other surfaces. Method of Assessment: Examination 100% Textbook: • Bondy J.A. and Murty U. S. R., Graph Theory with Applications, Macmillan, 1978. MAT2512 - Vector Analysis I Lecturers: Follows from: Leads to: Credit value: Lectures: Dr. J. Sultana MAT1511 MAT3504 4 20 20 Tutorials: Semester: • • • • • • • • • • 8 1 Double integrals; Triple integrals; Applications of multiple integrals; Change of variables in multiple integrals; Jacobian; Grad, div and curl operators; Space curves; Serret-Frenet formulae; Line integral and its applications; Conservative vector fields, scalar potential. Method of Assessment: Examination 100% Suggested Reading: • Finney R.L., Weir M.D. and Giordano F.R., Thomas Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001. • Camilleri C.J., Vector Analysis, Malta University Press, 1994 • Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997 MAT2513 - Vector Analysis II Lecturer : Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • • • • Dr. J. Sultana MAT2512 MAT3216 4 20 8 2 The Laplacian and other operators; Vector identities; Introduction to surfaces; Quadrics; Surface integrals; Applications of surface integrals; Green’s theorem; Gauss’ theorem; Stokes’ theorem; Orthogonal curvilinear coordinates. Method of Assessment: Examination 100% Suggested Reading: • Finney R.L., Weir M.D. and Giordano F.R., Thomas Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001. • Camilleri C.J., Vector Analysis, Malta University Press, 1994 • Roe J., Elementary Geometry, Oxford Science Publications, Clarendon Press, 1997 MAT2912 – Computational Mathematics Lecturers: Credit value: Lectures: Tutorials: Semesters: Various 2 10 4 1, 2 21 • Introduction to mathematical and computer algebra software; • Application to various branches of Mathematics. Method of Assessment: Examination 100% Suggested reading • Don E., Theory and Problems of Mathematica, Shaum’s Outline Series, McGraw-Hill, 2001. • Kreyszig E. and Normington E.J., Mathematica Computer Guide, John Wiley, 2002. • Wolfram S., Mathematica book, Wolfram Media, 5th Edition, 2003. Available online. Year III MAT3105 - Linear Algebra I This credit is identical to MAT2112 in the second year. Second and third year students attend together for MAT2112 and/or MAT3105, and they will have the same examination. After 2008, MAT3105 will be discontinued, and only MAT2112 will be given to the second years only. 22 Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • • • • • • • • • • • Prof. I. Sciriha MAT1101 MAT3905 4 20 8 1 Vector spaces; Dimension theorems; Third isomorphism theorem; Direct sum of spaces; Dual spaces; Inner product spaces; Gram Schmidt orthogonalisation; The rank of matrices; Projection of a vector space onto a subspace; Similar matrices; Eigenvalues and eigenvectors; The characteristic polynomial; The Cayley-Hamilton theorem; The minimum polynomial; Diagonalisation and applications; The Jordan normal form; Spectral decomposition. Method of Assessment: Examination 100% Textbooks • • • • • Lecture Notes. Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998. Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. Nering E.D., Linear Algebra and Matrix Theory, John Wiley and Sons, 2nd Edition, 1970. Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003. MAT3206 - Lebesgue Integration Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. D. Buhagiar MAT3792 MAT3218 4 20 8 2 23 • • • • • • algebras; Measure spaces; Lebesgue measure on R1 ; Measurable functions; Lebesgue integral; Convergence theorems; • The space Lp , 1 p . Method of Assessment: Examination 100% Suggested Reading: • Fremlin D.H., Measure Theory; Vol. 1: The Irreducible Minimum, Torres Fremlin, 2000. • Cohn D., Measure Theory, Birkhauser, 1980. • Bear H.S., A Primer of Lebesgue Integration, Academic Press, 2nd Edition, 2001. MAT3214 - Complex Analysis This credit will be offered for the first time in the third year, instead of the fourth year where it is offered as MAT3207. This credit will be discontinued from the fourth year after the academic year 2007-2008. Third year and fourth year students attend lectures together for MAT3214 and/or MAT3207. Lecturer: Follows from: Leads to: Credit value: Dr. J. Sultana MAT1211 MAT3218 4 24 Lectures: Tutorials: Semester : • • • • • • • • • • • • • • • • 20 8 1 Continuity and analytic functions; The Cauchy-Riemann equations; Exponential, trigonometric, hyperbolic and logarithmic functions; Harmonic functions; Contour integration; Fundamental theorem of calculus; Cauchy’s theorem; Cauchy’s integral formulae; Liouville’s theorem; The fundamental theorem of algebra; Sequences; Taylor’s series; Laurent’s series; Zeros and poles; Residues; Residue theorem and its applications. Method of Assessment: Examination 100% Suggested Reading • Osborne A.D., Complex Variables and their Applications, Addison-Wesley, New York, 1999. • Priestley H., Introduction to Complex Analysis, Oxford University Press, Oxford, 1994. MAT3504 - Vector Analysis IIA This credit will be offered for the last time in the academic year 2007-2008. In subsequent years, the subject matter of this credit will be eventually covered in MAT2513 and MAT3216. Lecturer : Follows from: Leads to: Credit value: Lectures: Dr. J. Sultana MAT2503 MAT3713 4 20 25 Tutorials: Semester: • • • • • 8 2 Scalar and vector fields: gradient, divergence, curl; Vector operators and identities; Line, surface and volume integrals; Integral theorems of Gauss, Green and Stokes; General orthogonal curvilinear coordinates. • The inverse function theorem; • The implicit function theorem. Method of Assessment: Examination 100% Suggested Reading • Finney R.L., Weir M.D. and Giordano F.R., Thomas’ Calculus, 10th Edition, Addison-Wesley Longman, New York, 2001. • Camilleri C.J., Vector Analysis, Malta University Press, Malta, 1994. • Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999. • Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976. • Apostol T., Calculus Vol. II: Calculus of Several Variables with applications to vector analysis, Blaisdell, 1961. • Thurston H., Intermediate Mathematical Analysis, Clarendon Press, Oxford, 1988. • Edwards C.H., Advanced Calculus of Several Variables, Academic Press, 1973. • Webb J.R.L., Functions of Several Variables, Ellis Horwood, 1991. MAT3602 - Mechanics Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. A. Buhagiar MAT1511, MAT1611 MAT3603 4 20 8 2 • Newton’s laws of motion: • Cartesian coordinates: 26 • Motion in a straight line and a plane, • Projectiles, • Simple harmonic motion; • Intrinsic coordinates: • The catenary, • The cycloid, • The isochronous pendulum; • Plane polar coordinates: • Central forces, • Binary systems, • Disturbed orbits and pedal coordinates; • Principle of conservation of energy: • Conservative forces, • Equivalent conditions, • Examples: • Orbital motion, • Motion on a surface under gravity; • Many-particle systems: • Motion of centroid, • Angular momentum and moment of force; • Rotational motion: • Rotation of a rigid body about a fixed axis, • Bodies rolling in a straight line; • Impulsive motion. Method of Assessment: Examination 100% Main Text • Lunn M., A first course in Mechanics, Oxford University Press, Oxford, 1991. Supplementary Reading: • Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969. • Camilleri C.J., Classical Mechanics, Malta 2004. MAT3701 - Differential Equations Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • Prof. A. Buhagiar MAT1211, MAT1090 MAT3712 4 20 8 2 Picard's theorem; Linear differential equations; Green’s function; Simple dynamical systems; 27 • First order quasi-linear equations. Method of Assessment: Examination 100% Suggested Reading • Burkill J.C., The Theory of Ordinary Differential Equations, Oliver & Boyd, 1962. • Arnold V.I., Ordinary Differential Equations, MIT Press, Cambridge, Massachusetts, 1978. • Hirsch M. and Smale S., Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, 1997. • Ayres F., Differential Equations, Schaum’s Outline Series, McGraw-Hill, 1972. • Tenebaum M. and Pollard H., Ordinary Differential Equations, Dover Publications, 1985. MAT3792 – Metric Spaces II and Uniform Convergence Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. D. Buhagiar, Dr. J. L. Borg MAT1211, MAT2292 MAT3206, MAT3218 4 20 8 1 • Metric spaces: • Compactness, • Connectedness, • Completeness. • Sequences and series of functions; 28 • Uniform convergence: • Continuity, • Differentiability, • Integration, • Applications. Method of Assessment: Examination 100% Textbook • Kolmogorov A. N. and Fomin S.V., Introductory Real Analysis, Dover, 1975. Suggested Reading • Sutherland W., Introduction to Metric and Topological spaces, Clarendon Press, Oxford, 1975. • Bryant V., Metric Spaces, Cambridge University Press, 1995. • Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974. • Spivak M., Calculus, Publish or Perish, 3rd Edition, 1994. • Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976. MAT3905 Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • - Linear Algebra II Prof. I. Sciriha MAT3105 MAT3218, MAT3414 2 10 4 1 Algebra of linear transformations; Dual spaces and annihilators; The transpose; The minimum polynomial; T-invariant spaces; Spectral decomposition; Criteria for diagonalisation; 29 • Jordan normal forms. Method of Assessment: Examination 100% Textbooks • Lecture Notes. • Kaye R. and Wilson R., Linear Algebra, Oxford Science Publications, Oxford, 1998. • Herstein I.N., Topics in Algebra, John Wiley & Sons, 3rd Edition, 1996. • Nering E.D., Linear Algebra and Matrix Theory, John Wiley and Sons, 2nd Edition, 1970. • Nicholson R., Linear Algebra with Applications, McGraw-Hill, 2003. Credits to be offered in the third year in and after 2008-2009. MAT3115 - Groups This credit will not be offered in 2007-2008. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. J. Lauri MAT1101 MAT3116 4 20 8 1 • Groups acting on finite sets: the orbit-stabiliser theorem; • Conjugacy; • Strong form of Cayley's theorem; 30 • • • • • Sylow’s theorems; Application to the classification of groups of low order; Automorphisms; Permutation groups; Burnside’s counting theorem. Method of Assessment: Examination 100% Main Text • Ledermann W. and Weir A.J., Introduction to Group Theory, Addison-Wesley Longman, 2nd Edition, 1996. Supplementary Reading • Armstrong M.A., Groups and Symmetry, Springer Verlag, Heidelberg, 1997. • Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 1989. • Herstein I.N., Topics in Algebra, Wiley, 3rd Edition, 1996. MAT3215 – Metric Spaces This credit will not be offered in 2007-2008. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • • Dr. D. Buhagiar MAT1211 MAT3216, MAT3217 4 20 8 1 Axioms and examples of metric spaces; Subspaces; Open and closed sets; Continuity; Equivalent metrics; Compactness and sequential compactness: their equivalence; Connectedness; 31 • Completeness. Method of Assessment: Examination 100% Textbook • Kolmogorov A. N. and Fomin S.V., Introductory Real Analysis, Dover, 1975. Suggested Reading • • • • Sutherland W., Introduction to Metric and Topological spaces, Clarendon Press, Oxford, 1975. Bryant V, Metric Spaces, Cambridge University Press, 1995. Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976. Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974. MAT3216 – Analysis IV This credit will not be offered in 2007-2008. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. J. L. Borg MAT2213 MAT3218, MAT3513 4 20 8 2 • Functions of several variables: • Directional derivatives and partial derivatives; • Vector-valued functions: • Differentiability and the total derivative; • The chain rule. 32 • The inverse and implicit function theorems. Method of Assessment: Examination 100% Suggested reading: • • • • Webb J.R.L., Functions of Several Variables, Ellis Horwood, 1991 Thurston H., Intermediate Mathematical Analysis, Clarendon Press, Oxford, 1988 Apostol T., Mathematical Analysis, Addison-Wesley, 2nd Edition, 1974 Rudin W., Principles of Mathematical Analysis, McGraw-Hill, 3rd Edition, 1976 MAT3413 – Probabilistic Combinatorics This credit will not be offered in 2007-2008. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • Dr. A. Vella MAT2412 MAT3414 2 10 4 1 The Monty Hall problem and the birthday paradox, Discrete sample spaces and discrete random variables, Bernoulli, binomial and geometric random variables, Random graphs, Simple lower bound for the Ramsey number R ( k , k ) ; 33 • • • • • Expectation and linearity of expectation, Coupon collector’s problem, Szele’s theorem on Hamiltonian paths, Existence of large cuts in graphs, Existence of large sum-free subsets within sets of positive integers; • • • • Randomized algorithms, Existence of large independent sets in graphs, Existence of graphs with large girth, Improved lower bound for R ( k , k ) ; • • • • Variance and standard deviation, Chebyshev’s inequality, Threshold functions for balanced subgraphs, Discussion of phase transition in random graphs; • • • • Conditional probability and independence, Erdos-Ko-Rado theorem, Lovasz local lemma, Applications: edge-disjoint paths in a graph, k - SAT, lower bound for the van der Waerden function; • Turan’s theorem. Method of Assessment: 80% Examination, 20% Oral Textbooks • Mitzenmacher M. and Upfal E., Probability and Computing. Randomized Algorithms and Probabilistic Analysis, Cambridge University Press, 2005. • Alon N. and Spencer J. H., The Probabilistic Method, John Wiley and Sons, 1992. Year IV MAT3116 – Group Representations Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • Prof. J. Lauri MAT1101, MAT3115 MAT5411 5 18 7 1, 2 Quick review of groups, vector spaces and linear transformations; Group representation over the complex field: matrix and module point of view; Maschke’s theorem and Schur’s lemma; reducibility and complete reducibility; Characters, inner products of characters and character tables; Some applications. 34 Method of Assessment: Examination 100% Main Text • James G. and Liebeck M., Representations and Characters of Groups, Cambridge University Press, Cambridge, 2001. Supplementary Reading • Serre J. P., Linear representations of Finite Groups, 4th Edition, Springer, 1977. • Nering E. D., Linear Algebra and Matrix Theory, 2nd Edition, Wiley, 1970. MAT3207 - Complex Analysis This credit will be given in the fourth year for the last time in 2007-2008. Subsequently it will be offered in the third year as MAT3214. Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters : • • • • • • • • Dr. J. Sultana MAT1211 MAT3218 5 18 7 1, 2 Continuity and analytic functions; The Cauchy-Riemann equations; Exponential, trigonometric, hyperbolic and logarithmic functions; Harmonic functions; Contour integration; Fundamental theorem of calculus; Cauchy’s theorem; Cauchy’s integral formulae; 35 • • • • • • • • Liouville’s theorem; The fundamental theorem of algebra; Sequences; Taylor’s series; Laurent’s series; Zeros and poles; Residues; Residue theorem and its applications. Method of Assessment: Examination 100% Textbooks • Osborne A.D., Complex Variables and their Applications, Addison-Wesley, New York, 1999. • Priestley H., Introduction to Complex Analysis, Oxford University Press, Oxford, 1994. MAT3219 – General Topology Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: • • • • Dr. D. Buhagiar MAT3792 MAT5211 5 18 7 1, 2 Cardinals and ordinals; Subspaces, product spaces and quotient spaces; Separation axioms and countability axioms; Connectedness, compactness and paracompactness. Method of Assessment: Examination 100% Main Text • Munkres J.R., Topology, Prentice Hall, 2nd Edition, 2000. 36 Supplementary Reading • Nagata J., Modern General Topology, North-Holland, 2nd revised edition, 1985. • Engelking R., General Topology, revised and completed edition, Helderman Verlag, 1989. MAT3513 – Tensors and Relativity Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: • • • • • • • • • • • • Dr. J. Sultana MAT3504 MAT5611 5 18 7 1, 2 Differentiable manifolds; Maps of manifolds; Tangent and cotangent spaces; Bases; Tensors and tensor algebra; The metric tensor; Tensor transformation law; Tensor fields; Christoffel symbols; Covariant derivative; Parallel propagation and geodesics; The Riemann tensor and its symmetries; 37 • • • • • • • • The Ricci tensor, curvature scalar and Weyl tensor; The Bianchi identities; Principle of equivalence; Gravitation as space-time curvature; Energy-momentum tensor; Perfect fluids; Einstein’s field equations; Schwarzschild black hole solution. Method of Assessment: Examination 100% Suggested Reading • Carroll S.M., Spacetime and Geometry: An Introduction to General Relativity, 1st Edition, Addison Wesley, New York, 2003. • Schutz B. F., A first course in General Relativity, Cambridge University Press, Cambridge, 1985. • Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999. MAT3613 - Classical Mechanics Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Prof. A. Buhagiar MAT1501, MAT3602 MAT5611, MAT5711 5 18 7 1, 2 • Rotating frames: • Angular velocity, • The rotating axes theorem, • Velocity and acceleration in a rotating frame, • The rotation of the earth; • Systems of many particles: • The two body problem, • Rigid bodies; • Rigid bodies: • Angular velocity and angular momentum, • The equation of motion, 38 • • • • The inertia tensor, Principal axes and principal moments of inertia, General rotation of a rigid body fixed at a point, Applications; • Lagrangian mechanics: • Generalised coordinates, • Virtual displacements, • Generalised forces, • Lagrange’s equations, • Ignorable coordinates; • Application of Lagrangian mechanics: • Rigid bodies, the Euler angles, • Precession of tops and rolling bodies, • Small oscillations, • Impulsive motion. Method of Assessment: Examination 100% Main Texts • Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991. Supplementary Reading • Camilleri C.J., Classical Mechanics, Malta, 2004. • Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969. • Marion J.B. and Thornton S.T., Classical Dynamics of Particles and Systems, 4th Edition, Harcourt College Publishers, New York, 1995. • Goldstein H., Classical Mechanics, 2nd Edition, Addison-Wesley, New York, 1980. MAT3713 - The Finite Element Method Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Prof. A. Buhagiar MAT3602, MAT3701 MAT5711 5 18 7 1, 2 • Approximate solution of boundary value problems: • Variational methods, • Weighted residual methods; • Variational methods: • The Rayleigh-Ritz method, • Minimisation of the energy functional, • Application to bars and beams, • Discretisation of region, • The piecewise Rayleigh-Ritz method, • One dimensional elements, • The element shape functions, • The linear bar element, • The cubic beam element, 39 • Element stiffness and consistent loading, • Assembly and solution of stiffness equations, • Estimate of accuracy of approximate solution; • Weighted residual methods: • The Galerkin method, • The one-dimensional Poisson equation: • Dirichlet, derivative or mixed boundary conditions, • Solution with linear or quadratic elements; • The two-dimensional Poisson equation: • Solution with linear triangular elements; • Applications of above: • Axial extension and vibration of bars, • Bending of beams, • Torsion, • Heat Transfer, • Groundwater flow. Method of Assessment: Examination 100%. Main Text • Lewis P. E. and Ward J. P., The Finite Element Method, Principles and Applications, Addison-Wesley, New York, 1991. Supplementary Reading • Dawe D.J., Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press, Oxford, 1984. • Segerlind L.J. Applied Finite Element Analysis, John Wiley, New York; 1st Edition 1976, 2nd Edition 1984. • Ottosen N. S. and Petersson H., Introduction to the Finite Element Method, Prentice Hall, New York, 1992. • Fagan M.J., Finite Element Analysis: Theory and Practice, Longman, Singapore, 1992. MAT3712 – Partial Differential Equations and Calculus of Variations This credit will not be offered in 2007-2008. Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Dr. J. Muscat MAT3701 MAT3613, MAT5711 5 18 7 1, 2 • Partial differential equations: • Separable solutions; • Elliptic equations: • Gravitation, • Laplace’s equation, • Poisson’s equation, • Harmonic functions; • Hyperbolic equations: • Waves, • D’Alembert solution; • Parabolic equations: • Heat, 40 • Fundamental solution. • Calculus of variations: • Motivating examples, • Continuous and piecewise differentiable solution, • The Euler-Lagrange equation, • Problems with fixed and non-fixed endpoints, • Constraints, • Necessary conditions for a minimum and corner conditions. Method of Assessment: Examination 100% Suggested Reading • Sneddon I., Elements of Partial Differential Equations, McGraw-Hill, 1957. • Haberman R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, New Jersey, 3rd Edition, 1998. • Farbon S., Partial Differential Equations for scientists and Engineers, Dover Publications, 1993. • Clegg J.C., Calculus of Variations, Oliver and Boyd, 1968. • Pars L.A., An Introduction to the Calculus of Variations, Heinemann, 1962. • Pinch E.R., Optimal Control and the Calculus of Variations, Oxford University Press, 1993. MAT3196 – Group Representations Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: • • • • • • Prof. J. Lauri MAT1101, MAT3115 MAT5411 6 22 8 1, 2 Quick review of groups, vector spaces and linear transformations; Group representation over the complex field: matrix and module point of view; Maschke’s theorem and Schur’s lemma; reducibility and complete reducibility; Characters, inner products of characters and character tables; Some applications; Further topics. Method of Assessment: Examination 100% Main Text • James G. and Liebeck M., Representations and Characters of Groups, Cambridge University Press, Cambridge, 2001. 41 Supplementary Reading • Serre J. P., Linear representations of Finite Groups, 4th Edition, Springer, 1977. • Nering E. D., Linear Algebra and Matrix Theory, 2nd Edition, Wiley, 1970. MAT3299 – General Topology Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: • • • • • Dr. D. Buhagiar MAT3792 MAT5211 6 22 8 1, 2 Cardinals and ordinals; Subspaces, product spaces and quotient spaces; Separation axioms and countability axioms; Connectedness, compactness and paracompactness; Further topics. Method of Assessment: Examination 100% Main Text • Munkres J.R., Topology, Prentice Hall, 2nd Edition, 2000. Supplementary Reading • Nagata J., Modern General Topology, North-Holland, 2nd revised edition, 1985. • Engelking R., General Topology, revised and completed edition, Helderman Verlag, 1989. 42 MAT3593 – Tensors and Relativity Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: • • • • • • • • • • • • • • • • • • Dr. J. Sultana MAT3504 MAT5611 6 22 8 1, 2 Differentiable manifolds; Maps of manifolds; Tangent and cotangent spaces; Bases; Tensors and tensor algebra; The metric tensor; Tensor transformation law; Tensor fields; Christoffel symbols; Covariant derivative; Parallel propagation and geodesics; The Riemann tensor and its symmetries; The Ricci tensor, curvature scalar and Weyl tensor; The Bianchi identities; Principle of equivalence; Gravitation as space-time curvature; Energy-momentum tensor; Perfect fluids; 43 • Einstein’s field equations; • Schwarzschild black hole solution; • Further topics. Method of Assessment: Examination 100% Suggested Reading • Carroll S.M., Spacetime and Geometry: An Introduction to General Relativity, 1st Edition, Addison Wesley, New York, 2003. • Schutz B. F., A first course in General Relativity, Cambridge University Press, Cambridge, 1985. • Camilleri C.J., Tensor Analysis, Malta University Press, Malta, 1999. MAT3693 - Classical Mechanics Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Prof. A. Buhagiar MAT1501, MAT3602 MAT5611, MAT5711 6 22 8 1, 2 • Rotating frames: • Angular velocity, • The rotating axes theorem, • Velocity and acceleration in a rotating frame, • The rotation of the earth; • Systems of many particles: • The two body problem, • Rigid bodies; • Rigid bodies: • Angular velocity and angular momentum, • The equation of motion, • The inertia tensor, • Principal axes and principal moments of inertia, • General rotation of a rigid body fixed at a point, • Applications; • Lagrangian mechanics: 44 • • • • • Generalised coordinates, Virtual displacements, Generalised forces, Lagrange’s equations, Ignorable coordinates; • Application of Lagrangian mechanics: • Rigid bodies, the Euler angles, • Precession of tops and rolling bodies, • Small oscillations, • Impulsive motion; • Further topics. Method of Assessment: Examination 100% Main Texts • Lunn M., A first Course in Mechanics, Oxford University Press, Oxford, 1991. Supplementary Reading • Camilleri C.J., Classical Mechanics, Malta, 2004. • Charlton F., Textbook of Dynamics, van Nostrand Company Ltd., London, 1969. • Marion J.B. and Thornton S.T., Classical Dynamics of Particles and Systems, 4th Edition, Harcourt College Publishers, New York, 1995. • Goldstein H., Classical Mechanics, 2nd Edition, Addison-Wesley, New York, 1980. MAT3793 - The Finite Element Method Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Prof. A. Buhagiar MAT3602, MAT3701 MAT5711 6 22 8 1, 2 • Approximate solution of boundary value problems: • Variational methods, • Weighted residual methods; • Variational methods: • The Rayleigh-Ritz method, • Minimisation of the energy functional, • Application to bars and beams, • Discretisation of region, • The piecewise Rayleigh-Ritz method, • One dimensional elements, • The element shape functions, • The linear bar element, • The cubic beam element, • Element stiffness and consistent loading, • Assembly and solution of stiffness equations, • Estimate of accuracy of approximate solution; • Weighted residual methods: • The Galerkin method, • The one-dimensional Poisson equation: 45 • Dirichlet or derivative boundary conditions, • Solution with linear or quadratic elements; • The two-dimensional Poisson equation: • Solution with linear triangular elements; • Applications of above: • Axial extension and vibration of bars, • Bending of beams, • Torsion, • Heat Transfer, • Groundwater flow; • Further topics. Method of Assessment: Examination 100%. Main Text • Lewis P. E. and Ward J. P., The Finite Element Method, Principles and Applications, Addison-Wesley, New York, 1991. Supplementary Reading • Dawe D.J., Matrix and Finite Element Displacement Analysis of Structures, Clarendon Press, Oxford, 1984. • Segerlind L.J. Applied Finite Element Analysis, John Wiley, New York; 1st Edition 1976, 2nd Edition 1984. • Ottosen N. S. and Petersson H., Introduction to the Finite Element Method, Prentice Hall, New York, 1992. • Fagan M.J., Finite Element Analysis: Theory and Practice, Longman, Singapore, 1992. MAT3782 – Partial Differential Equations and Calculus of Variations This credit will not be offered in 2007-2008. Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Dr. J. Muscat MAT3701 MAT3613, MAT5711 6 22 8 1, 2 • Partial differential equations: • Separable solutions; • Elliptic equations: • Gravitation, • Laplace’s equation, • Poisson’s equation, • Harmonic functions; • Hyperbolic equations: • Waves, • D’Alembert solution; • Parabolic equations: • Heat, • Fundamental solution. • Calculus of variations: • Motivating examples, 46 • • • • • Continuous and piecewise differentiable solution, The Euler-Lagrange equation, Problems with fixed and non-fixed endpoints, Constraints, Necessary conditions for a minimum and corner conditions. • Further topics. Method of Assessment: Examination 100% Suggested Reading • Sneddon I., Elements of Partial Differential Equations, McGraw-Hill, 1957. • Haberman R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, New Jersey, 3rd Edition, 1998. • Farbon S., Partial Differential Equations for scientists and Engineers, Dover Publications, 1993. • Clegg J.C., Calculus of Variations, Oliver and Boyd, 1968. • Pars L.A., An Introduction to the Calculus of Variations, Heinemann, 1962. • Pinch E.R., Optimal Control and the Calculus of Variations, Oxford University Press, 1993. THE PROJECT MODULES For the project modules, students choose either MAT3218 or MAT3414 (12 credits), together with MAT3990, the Mathematics Seminar (6 credits), for a total of 18 credits. These credits are described next. MAT3218 - Functional Analysis Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Dr. D. Buhagiar, Dr. J. Hamhalter MAT3206 MAT5211 12 45 15 1, 2 • • Metric spaces and their completion; Normed vector spaces, Banach spaces; • • • • Examples of normed spaces: the spaces Bounded linear operators, dual spaces; Hilbert spaces; Orthogonal projections. • • • • • Hahn-Banach theorem; Uniform boundedness theorem; Open mapping theorem, closed graph theorem; Spectral theory; Compact and self adjoint operators. l p , Lp and C[a, b] ; 47 Method of Assessment: Examination 100% Main text • Kreysig E., Introductory Functional Analysis, Wiley, 1989. Supplementary Reading • Rudin W., Functional Analysis, Tata McGraw-Hill, 1973. • Kolmogorov A.N. and Fomin S.V., Elements of the Theory of Functions and Functional Analysis, Dover, 1957. MAT3414 - Discrete Mathematics Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semesters: Prof. J. Lauri, Prof. I. Sciriha. MAT2412 MAT5411 12 45 15 1, 2 • • • • • • Definitions and elementary results on graphs; Trees; Connectivity; Euler tours and Hamilton cycles; The cycle index of a permutation group and the use of Polya’s theorem; Ramsey’s theorem for graphs; • • • • • • Vector spaces associated with graphs; Cycle-cutset duality; Graph colourings; Planar graphs; Introduction to error-correcting codes; Introduction to combinatorial designs. Method of Assessment: Examination 100% Main Texts • Wilson R.J., Introduction to Graph Theory, Longman, 4th Edition, 1996. • West D.B., Introduction to Graph Theory, Prentice Hall, 2nd Edition, 2001. • Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, 1989. 48 Supplementary Reading • Cameron P.J., Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994. • Bryant V., Aspects of Combinatorics: A Wide Ranging Introduction, Cambridge University Press, Cambridge, 1993. MAT3990 – Lecturers: Credit value: Semesters: • Mathematics Seminar Various 6 1, 2 In this study unit, a long essay and a seminar are presented by the student on a suitable topic in Mathematics. Method of Assessment: Essay/Seminar 100% FACULTIES OF ENGINEERING & ARCHITECTURE Year I MAT1801 – Mathematics for Engineers I Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Dr. J. L. Borg, Dr. A. Vella, Mr. J. B. Gauci A-Level MAT1802 4 20 8 1 • • • Matrices and determinants; Systems of linear equations; Matrices: eigenvalues and eigenvectors. • • • First order differential equations; Second order linear differential equations with constant coefficients; Partial differentiation. Method of Assessment: Examination 100% Textbooks • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. • Spiegel M.R., Advanced Calculus, Schaum’s Outline Series, McGraw-Hill, 1981. • Finney R.L., Weir M.D., Giordano F.R., Thomas’ Calculus, Addison-Wesley Longman, New York, 10th Edition, 2001. MAT1802 - Mathematics For Engineers II Lecturer: Dr. J. L. Borg, Dr. A. Vella, Mr. J. B. Gauci 49 Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: A-level, MAT1801 MAT2803, MAT2804 4 20 8 2 • • • Vector algebra and introduction to vector spaces; Transformation of rectangular Cartesian coordinates on a plane and in space; linear transformations; Linear objects: lines (in 2D and 3D), planes (in 3D). • • • Double integrals; Fourier series; Sequences and series Method of Assessment: Examination 100% Textbooks • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. • Spiegel M.R., Advanced Calculus, Schaum’s Outline Series, McGraw-Hill, 1981. • Finney R.L., Weir M.D., Giordano F.R., Thomas’ Calculus, Addison-Wesley Longman, New York, 10th Edition, 2001. Year II MAT2803 - Laplace and Fourier Transforms Lecturer: Follows from: Credit value: Lectures: Tutorials: Semester: Dr. J. L. Borg MAT1802 2 10 4 1 • Laplace transforms: • The delta function and the unit step function, • Transforms of elementary functions, • Properties including linearity, scaling, shift, modulation, convolution and correlation, • Transforms of integrals and derivatives, • Differential equations; solution of initial and boundary value problems, • Impulse response and the transfer function; • Fourier transforms: • Fourier’s identity, • Transform pairs, duality and symmetry, • Properties including linearity, scaling, shift, modulation, convolution and correlation, • Rayleigh’s theorem and the power theorem, • Transforms of derivatives, • Solution of boundary value problems. Method of Assessment: Examination 100% Main Text • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. Supplementary Reading • Spiegel M.R., Laplace Transforms, Schaum’s Outline Series, McGraw-Hill, New York, 1994. • Senior T.B., Mathematical Methods in Electrical Engineering, Cambridge University Press, Cambridge, 50 1986. • Bracewell R.N., The Fourier Transform and its Applications, 3rd Edition, McGraw-Hill, New York, 2000. MAT2804 - The Eigenvalue Problem and Multiple Integrals Lecturer: Follows from: Credit value: Lectures: Tutorials: Semester: Dr. J. L. Borg MAT1801, MAT1802 2 10 4 2 • Eigenvalues and eigenvectors with application to simultaneous linear differential equations; • Transformations of double and triple integrals. Method of Assessment: Examination 100% Textbook • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. Year III MAT3805 – Optimisation, Complex & Numerical Analysis I Lecturer: Follows from: Credit value: Lectures: Tutorials: Semester: Dr. J. Sultana, Prof. A. Buhagiar MAT1801, MAT1802 4 20 8 1 • Optimisation: • Local and global extrema of functions of several variables, • Lagrange’s method for constrained problems, • Lagrange’s method with two constraints; • Functions of a complex variable: • Notation and definitions, • Limits, continuity and analytic functions, • The Cauchy-Riemann equations, • Exponential, trigonometric, hyperbolic, logarithmic and power functions, • Harmonic functions. • Numerical Analysis: • Locating roots of equations: • The Newton Raphson in one and two dimensions: rate of convergence, • The variable secant method, • The fixed point theorem, x = f (x ), • The method of steepest descent, • Bairstow’s method for the quadratic factors of a polynomial with real coefficients, • The quotient difference method for the roots of a polynomial; • Solution of linear equations: • Gaussian elimination, • Cholesky’s LU and LLT methods, • Iterative methods: Jacobi, Gauss-Seidel and the SOR methods; 51 • Eigenvalues and eigenvectors: • Bounds for eigenvalues using Gershgorin’s theorem, • The power method for the largest and smallest eigenvalues, and the corresponding eigenvectors, • Rayleigh’s quotient for Hermitean matrices, • Jacobi’s method of rotations for real symmetric matrices, • Applications to quadratic forms and normal modes. Method of Assessment: Examination 100% Textbooks • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett Publishers, 3rd Edition, 2006. • Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, New York, 1997. Supplementary Reading • Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling W.T., Numerical Recipes in Fortran, Cambridge University Press, Cambridge, 1989. • Rajasekaran S., Numerical Methods in Science and Engineering, 2nd Edition, Wheeler Publishing, New Delhi, 1999. • Burden R. L. and Faires J. D., Numerical Analysis, 7th Edition, Brooks Cole, London, 2001. • Cheney E. W. and Kincaid D. R., Numerical Mathematics and Computing, 4th Edition, Brooks Cole, 1999. MAT3806 - Optimisation, Complex & Numerical Analysis II Lecturer: Follows from: Credit value: Lectures: Tutorials: Semester: Dr. J. Sultana, Prof. A. Buhagiar MAT1801, MAT1802 4 20 8 2 • Functions of a complex variable: • • • • • • • • • Contour integration, Fundamental theorem of calculus, Cauchy’s theorem, Cauchy’s integral formulae, Sequences and series, Taylor’s series, Laurent’s series, Zeros and poles, Residues. • Numerical analysis: • Interpolation: • Lagrangian interpolation, • The divided difference table, • The difference table and the Newton Gregory forward polynomial, • Inverse interpolation, • Spline interpolation; • Numerical differentiation: • Central difference formulae for the first and second derivatives, • Forward difference formulae for the first derivative, • Improvement by extrapolation; • Numerical integration: • Trapezoidal rule and Simpson’s 1/3 and 3/8 rules, • Errors for the local and global versions of these rules, • Gaussian quadrature; 52 • Ordinary differential equations: • Euler’s method, • The modified Euler method, • The Runge-Kutta method; • The finite difference method for partial differential equations: • Laplace’s equation, • Poisson’s equation, • The transient heat equation. Method of Assessment: Examination 100% Textbooks • Zill D.G. and Cullen M.R., Advanced Engineering Mathematics, Jones and Bartlett, 3rd Edition, 2006. • Gerald C.F. and Wheatley P.O., Applied Numerical Analysis, 6th Edition, Addison-Wesley, 1997. Supplementary Reading • Press W.H., Flannery B.P., et al., Numerical Recipes in Fortran, Cambridge University Press, 1989. • Rajasekaran S., Numerical Methods in Science and Engineering, 2nd Edition, Wheeler Publishing, 1999. • Burden R. L. and Faires J. D., Numerical Analysis, 7th Edition, Brooks Cole, London, 2001. • Cheney E. W. and Kincaid D. R., Numerical Mathematics and Computing, 4th Edition, Brooks Cole, 1999. FACULTY OF ECONOMICS, MANAGEMENT & ACCOUNTANCY Year I MAT1991 - Mathematics I and II Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. A. Buhagiar, Dr. A. Vella, Mr. J. Sciriha Intermediate Mathematics MAT1902 8 40 16 1, 2 This credit comprises the two study units MAT1901 and MAT1902 which are described below. Students cannot take MAT1991 with MAT1901 and / or MAT1902. MAT1901 - Mathematics I Lecturers: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. A. Buhagiar, Dr. A. Vella, Mr. J. Sciriha Intermediate Mathematics MAT1902 4 20 8 1 • Rectangular Cartesian coordinates; • Linear equalities and inequalities: • The straight line, • Systems of linear equalities and inequalities and their representation on the plane, • Introduction to linear programming in two variables; • Functions: 53 • The linear, quadratic, exponential and logarithmic functions and their graphs, • Functions related to business: cost, revenue, profit, demand and supply functions; • Matrices: • Basic properties, • Elementary row operations, • Determinants, • The matrix inverse, • Systems of linear equations, • Easy applications; • Differential calculus: • Differentiation of simple algebraic, exponential and logarithmic functions, • Maxima and minima, • Applications to business problems. Method of Assessment: Examination 100% Recommended Text • Barnett R.A. and Ziegler M.R., College Mathematics, Prentice Hall, New York, 1996. MAT1902 - Mathematics II Lecturers: Follows from: Credit value: Lectures: Tutorials: Semester: Dr. A. Vella, Prof. A. Buhagiar, Mr. J. Sciriha MAT1901 4 20 8 2 • Mathematics of finance: • Simple and compound Interest, • Present and future value, • Effective rate of Interest, • Annuities and sinking funds; • Integration: • Integral calculus, • Antiderivatives, • Integration by substitution, • Integration by parts, • The definite integral, • Calculation of areas, • The trapezoidal rule, • Applications like consumers’ surplus and the Lorenz curve, • Solution of differential equations using separation of variables; • Matrices: • The input-output model, • The brand-switching model; • Functions of two variables: • Partial derivatives, • Maxima, minima and saddle points, • Optimisation with constraints, Lagrange multipliers, • Applications to problems in business and economics. Method of Assessment: Examination 100% 54 Recommended Text • Barnett R.A. and Ziegler M.R., College Mathematics, Prentice Hall, New York, 1996. BOARD OF STUDIES FOR IINFORMATION TECHNOLOGY Year I MAT1803 - Mathematical Transform Techniques Lecturer: Follows from: Credit value: Lectures: Tutorials: Semester: Mr. J. Borg MAT1001, MAT1002 2 10 4 2 • Laplace transforms; • Fourier series; • Fourier transforms. Method of Assessment: Examination 100% Textbooks • Spiegel M.R., Laplace Transforms, Schaum’s Outline Series, McGraw-Hill, New York, 1994. • Senior T.B., Mathematical Methods in Electrical Engineering, Cambridge University Press, Cambridge, 1986. MAT2402 - Networks Lecturer: Follows from: Leads to: Credit value: Lectures: Tutorials: Semester: Prof. J. Lauri A-Level MAT3490 2 10 4 1 • Basic graph theoretical concepts; • Eigenvector methods for ranking vertices in a graph; • Shortest paths and minimum spanning tree; 55 • • • • • Flow in networks, max-flow-min-cut theorem; Matching in bipartite graphs; Critical path and PERT techniques; Scheduling and sequencing problems; Travelling salesman problem. Method of Assessment: Examination 100% Main Text: One of • Dolan A., Aldous J., Networks and Algorithms, An Introductory Approach, John Wiley & Sons, 1994. or • Dossey J. A., Otto A. D., Spence L.E. and Eynden C.V., Discrete Mathematics, Addison–Wesley, 5th Edition, 2006. Supplementary reading • Biggs N.L., Discrete Mathematics, Oxford Science Publications, Clarendon Press, Oxford, 1989. • Gibbons A., Algorithmic Graph Theory, Cambridge University Press, 1985. • Wilson R.J., Introduction to Graph Theory, Longman, 4th Edition, 1996. 56
